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Power Up

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x


This is a beautiful result involving a parabola and parallels.


In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

Spot the Difference

Age 16 to 18 Short Challenge Level:

In this problem there are three equations of straight lines and three more complicated relations between x and y.

Consider all the factors of the non-linear expressions to discover what points lie on their graphs.

The more complicated relations have extra points (x, y) which satisfy the relations but the computer does not notice.You have to be cleverer than the computer and find the extra points satisfying the relation that do not show up on the computer's graph. To do this you need to look at all the factors in the expressions and find when they are zero.